Executive Summary : | PI's research interests are mainly in the area of inverse problems and their applications to imaging sciences. More precisely, the focus is on inversion methods and reconstruction algorithms for certain integral transforms, the study of singularities using microlocal techniques, and coefficient recovery problems for partial differential equations. The research on such problems has been very active in the last few decades with a wide range of applications, notably including acoustic ow imaging using time-of-flight measurements, geophysics, reconstruction of velocity vector fields in blood vessels, photoelasticity, and many more. Below, we give a brief introduction for such problems: The focus is on questions like injectivity, inversion, and support theorems for a special class of integral transforms. In simple words, the inverse problem of interest is to recover a function (or more generally a tensor field) from the knowledge of its weighted integrals along with a collection of straight lines. These problems usually appear as a crucial step in several non-invasive approaches to imaging problems that have applications in medical imaging, seismology, ocean imaging, and many more. The proposer has contributed with different inversion methods (explicit inversion and microlocal inversion), support theorems, and injectivity results for certain integral transforms. Here are typical questions asked related to the integral transforms under consideration: 1. Is the integral transform injective modulo the kernel? 2. How to explicitly and efficiently invert the integral transform? 3. How to characterize the range of the operator? 4. Determine the extent to which the wavefront set of a symmetric m-tensor field can be recovered if the wavefront set of its integral transform is known? The proposed objectives in this proposal will address some of these questions for certain integral transforms (coming from some physical motivation) in various settings. |